Eu . Phys. J. Plus (2024) 139:489
h ps://doi.o g/10.1140/epjp/s13360-024-05310-z
Regula A icle
A s udy o sel -adjoin ness, Lie analysis, wa e s uc u es, and conse a ion laws
o he comple ely gene alized shallow wa e equa ion
Ali R. Ansa i1, Adil Jhangee 2,3, Mudassa Im an4, Beenish5, Mus a a Inc6,a
1Cen e o Applied Ma hema ics and Bioin o ma ics (CAMB), Gul Uni e si y o Science and Technology, Muba ak Al-Abdullah, Kuwai
2IT4Inno a ions, VSB – Technical Uni e si y o Os a a, Os a a-Po uba, Czech Republic
3Depa men o Ma hema ics, Namal Uni e si y, 30 KM Talagang Road, Mianwali 42250, Pakis an
4College o Humani ies and Sciences, Ajman Uni e si y, Ajman, UAE
5Depa men o Ma hema ics, Quaid-I-Azam Uni e si y, Islamabad 45320, Pakis an
6Depa men o Ma hema ics, Fi a Uni e si y, 23119 Elazig, Tü kiye
Recei ed: 14 Oc obe 2023 / Accep ed: 24 May 2024
© The Au ho (s) 2024
Abs ac This a icle explo es he analysis o he comple ely gene alized Hi o a–Sa suma–I o equa ion h ough Lie symme y
analysis. The equa ion unde conside a ion ep esen s a mo e comp ehensi e o m o he (2+1)-dimensional HSI equa ion, encom-
passing ou addi ional second-o de de i a i e e ms: 3H,4Hι,3H,4Hι,and6Hιι, eme ging om he inclusion
o second-o de dissipa i e- ype elemen s. We calcula e he in ini esimal gene a o s and de e mine he symme y g oup o each
gene a o using he Lie g oup in a iance condi ion. Employing he conjugacy classes o he Abelian algeb a, we ans o m he consid-
e ed equa ion in o an o dina y di e en ial equa ion h ough simila i y educ ion. Subsequen ly, we sol e hese o dina y di e en ial
equa ions o de i e closed- o m solu ions o he comple ely gene alized Hi o a–Sa suma–I o equa ion unde ce ain condi ions. Fo
o he scena ios, we u ilize he ex ended di ec algeb aic me hod o ob ain soli on solu ions. Fu he mo e, we igo ously calcula ed
he conse ed quan i ies co esponding o each symme y gene a o , he conse a ion laws o he model a e es ablished using he
mul iplie app oach. Addi ionally, we p esen he g aphical ep esen a ion o selec ed solu ions o speci ic alues o he physical
pa ame e s o he equa ion unde sc u iny.
1 In oduc ion
The mos accu a e ma hema ical ep esen a ions o na u al phenomena o en hinge on di e en ial equa ions, encompassing bo h
linea and nonlinea o mula ions. Nonlinea sys ems, in pa icula , equen ly yield such equa ions du ing he modeling p ocess,
nea ly all ields o con empo a y science, spanning om plasma physics, a omic physics, and solid-s a e physics o as onomy,
ma hema ics, biology, and chemis y, engage in in es iga ions conce ning hese nonlinea e olu ion equa ions. In his a icle, we
del e in o he examina ion o he (2+1)-dimensional HSI equa ion, augmen ed by he inco po a ion o ou addi ional second-o de
de i a i e e ms:
Hι +3(HHι +HιH)+6Hιι +1Hυι +5Hυυ +4Hι +3Hυ +2H 0.(1)
The equa ion discussed abo e is ecognized as he comple ely gene alized Hi o a–Sa suma–I o (cgHSI) Eq. [1], as i encompasses
e e y second-o de dissipa i e- ype elemen . The cgHSI equa ion se es as a model o shallow wa e wa es, delinea ing he
ampli ude o hese wa es h ough he unc ion H, which elies on he componen s ,υ,andι. I is exp essed using pa ial
de i a i es conce ning he spa ial componen s and υ, as well as he empo al componen ι. The equa ion inco po a es ee
pa ame e s deno ed by 1,2,3,4,5,and6. Shallow wa e equa ions o mo ion a e employed o depic he ho izon al
s uc u e o an a mosphe e and he dynamics o shallow wa e wa es, po aying he beha io o an incomp essible luid unde he
in luences o o a ional and g a i a ional accele a ions.
Se e al signi ican ad ancemen s ha e been achie ed in he esea ch o he (2+1)-gene alized shallow wa e equa ion and i s
a ian s. No ably, ou ully gene alized shallow wa e model s ands ou uniquely in he ealm o esea ch publica ions. We ha e
no iced a ple ho a o s udies delinea ing dis inc wa e s uc u es unde he same model name. In Gao and Tian [2], he (2+1)-
dimensional gene alized shallow wa e wa e equa ion is shown using he echnique o he gene alized anh me hod wi h symbolic
compu a ion, exhibi ing he s uc u e ha ollows:
Hυι +Hυ −3H Hυ−3HHυ 0.
ae-mail: [email p o ec ed] (co esponding au ho )
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489 Page 2 o 18 Eu . Phys. J. Plus (2024) 139:489
In his esea ch pape [3], p ecise solu ions and conse ed ec o s o he wo-dimensional gene alized shallow wa e wa e equa ion,
cha ac e ized by he subsequen s uc u e, a e elucida ed:
Hι +1HυH +21HHυ +2Hυ +3Hυ 0.
He e, 1and 2a e seen as some hing ha isn’ ze o, ega dless o whe he i is posi i e o nega i e. The pe iodic wa e solu ion o
he gene alized shallow wa e wa e equa ion [4] is ob ained using he imp o ed Jacobi ellip ic unc ion me hod wi h he ollowing
s uc u e:
Hι +1HHι +2HιH −Hι −3H 0,
whe e 1,2,3a e a bi a y nonze o cons an s. By employing he homogeneous balance me hod, an au o-Bäcklund ans o -
ma ion o he gene alized shallow wa e wa e equa ion [5] is de i ed wi h he ollowing s uc u e:
Hι +1HHι +2HιH −Hι −H 0.
In his equa ion, 1and 2 ep esen a bi a y nonze o cons an s. In he ci ed wo k [6], he au ho s del e in o (2 + 1)-dimensional
gene alized Hi o a–Sa suma–I o equa ions, p o iding insigh s in o Lie symme y analysis, in a ian solu ions, and dynamics o
soli on solu ions o he ollowing model:
ι+Hι +3(Hw)+1H0,
Hυ ,Hιw.
He e, His a physical ield. and windica o s o po en ials o ield de i a i es physical ield. Alongside his, and υ e e o
dimensions o space, whe eas ιis he o he ime a iable. Abundan exac in e ac ion solu ions, such as lump-soli on, lump-kink,
and lump-pe iodic solu ions, a e compu ed o he (2 + 1)-dimensional Hi o a–Sa suma–I o equa ion desc ibed [7], ollowing his
model:
HιHι +3HH
ι−3H ι−H, H.
Resea ch in o shallow-wa e wa e phenomena wi hin a (2+1)-dimensional Hi o a–Sa suma–I o sys em has led o in es iga ions on
X- ype soli on, esonan Y- ype soli on, and hyb id solu ions [8]:
HιHι +3HH
ι−3H ι−H, −H.
Resea che s in di e en disciplines ha e in es iga ed a ious echniques o sol ing PDE, many o which p oduce he soli a y
wa e solu ion. E ec i e s a egies o add essing hese nonlinea di e en ial p oblems include he ex ended di ec algeb aic me hod
[9], he gene alized exp(−φ(ζ)) expansion echnique [10], he symme y me hod [11], he anh-co h unc ion me hod [12], he
sine-cosine echnique [13], he exp- unc ion me hod [14], he new ex ended auxilia y equa ion me hod [15], he new Jacobi ellip ic
unc ions me hod [16], he new ex ended gene alized Kud yasho scheme [17], he esidual powe se ies me hod [18]. In addi ion,
app oxima e analy ical and nume ical me hods such as he homo opy analysis me hod (HAM) [19], he Adomian decomposi ion
me hod (ADM) [20], he a ia ional i e a ion me hod (VIM) [21], and he a ia ional homo opy pe u ba ion me hod (VHPM) [22].
One o he mos app op ia e analy ical app oaches o educe he complexi y o nonlinea pa ial di e en ial equa ions (PDE),
p oposed by No wegian ma hema ician Ma cus Sophus Lie (1842–1899), is he symme y me hod [23]. The cen e piece o he Lie
symme y me hod is he exis ence o ans o ma ion which makes solu ions o he di e en ial equa ion we conside , ixed poin s.
Lie ec o ields a e ound h ough his echnique o wha is unde discussion in di e en ial equa ions o be in a ian . The o e lying
g oup o symme y is he basis o he solu ion wi h an in a ian g oup o simila i y o he PDE. As a esul o imp o ing me hods
o con inuous Lie g oups s udy, bi u ca ion heo y, con ol heo y, classical mechanics, and ela i i y ha e been gi en new ools ha
helped in es ablishing he ole o symme y in di e en b anches o science [24].
The p inciple o conse a ion s a es ha he amoun o a sys em o physical en i ies does no dec ease as ime goes on, hus,
i is conse ed. Fo physical sys ems ha a e go e ned by he laws o mo ion, he equa ions o mo ion a e usually de i ed om
conse a ion laws such as he p inciple o conse a ion o momen um. The collec ion may also p ese e o he physical quan i ies
ha a e wo h knowing abou he physical p ocess con olling he mo ion. On op o ha , he conse a ion law knowledge will be
use ul o sol ing me hods, o ins ance, conse a i e nume ical me hods ha assume he shape o conse a ion law [25].
Noe he was i s o poin ou his c ucial and de ining cha ac e is ic in 1918 and she p opounded he Noe he heo em which is a
ma hema ical explana ion o his ela ionship be ween symme y and conse a ion laws [26]. Subsequen ly, nume ous esea che s
ha e endea o ed o de ise no el app oaches o de e mining conse a ion laws [27]. Add essing a majo limi a ion o his inding
he absence o a Lag angian Ib agimo ex ended he Noe he heo em [28]. Resea che s commonly employ he concep o nonlinea
sel -adjoin ness and he o mula ion o local conse a ion laws o es ablish conse a ion laws o di e en ial equa ions lacking a
classical Lag angian. A selec ion o hei wo ks is p esen ed in [29,30]. In his s udy, h ough me iculous analysis o he Lie g oup
and new auxilia y equa ion g oups associa ed wi h he model unde in es iga ion, we ha e gene a ed a ious pa e ns o soli a y
wa e solu ions.
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The s uc u e o he pape is ou lined as ollows: Sec ion (2) in oduces he ounda ional concep s o Lag angian and nonlinea
sel -adjoin ness. Sec ion 3conduc s a Lie analysis o iden i y he symme ies o Eq. (1) and employs he new auxilia y equa ion
me hod o de e mine he a eling wa e s uc u e o Eq. (1). Addi ionally, his sec ion o e s a g aphical ep esen a ion o some
pe inen solu ions o Eq. (1). Sec ion 4is dedica ed o nonlinea sel -adjoin ness analysis and he compu a ion o conse ed ec o s
o Eq. (1). Finally, concluding ema ks a e p o ided, summa izing he esea ch indings and b inging closu e o he pape .
2 P elimina ies
Suppose a sys em o nPDEs o p h o de is de ined as:
GγGγ(,H,H1,H2,...,Hp)0, γ1, 2, 3, ...,n,(2)
whe e nand q ep esen dependen and independen a iables espec i ely, and H1,H2,...,Hndeno e he i s , second, up o he
n h o de pa ial de i a i es o H. The o al di e en ial ope a o co esponding o kis:
Ak∂k+Hγ
k∂Hγ+Hγ
kl ∂Hγ
l+Hγ
klm∂Hγ
lm +...,k1, 2, ...q,(3)
whe e ∂k∂
∂k,∂Hγ∂
∂Hγ,∂Hγ
l∂
∂Hγ
l
,∂Hγ
lm ∂
∂Hγ
lm
.
2.1 Fo mal Lag angian
The o mal Lag angian o Eq. (2) can be de ined as ollows:
S
n

β1
hβSβ.(4)
Sis known as he o mal Lag angian, whe e h(h1,h2,...,hn) ep esen s he dependen a iables.
The adjoin o mula u ns in o his:
G∗
γ(,H,h,H1,h1,H2,h2,H2,....,HP,hp)δ(S)
δHγ0.(5)
The a ia ional de i a i e is de ined as:
δ
δHγ∂γ
H+∞

g1
(−1)gAk1...Akg∂Hγ
k1...kg
,γ1, 2, 3, ....,n.(6)
2.2 Nonlinea sel -adjoin ness
De ini ion 1 Equa ion (1) is said o be s ic ly sel -adjoin i he equa ion ob ained by subs i u ing ϒH om i s adjoin equa ion
is iden ical o he o iginal equa ion Eq. (2):
G∗
γϒHϒ(,H,...)Gγ.
De ini ion 2 The equa ion exp essed in Eq. (1) is said o be quasi-sel -adjoin i , upon subs i u ing ϒω(H), whe e ω(H) 0,
om i s adjoin equa ion, he esul ing equa ion is iden ical o he one in Eq. (2):
G∗
γϒω(H)ϒ(,H,...)Gγ.
De ini ion 3 Equa ion (1) is said o be weak sel -adjoin i he equa ion gene a ed by subs i u ing ϒω(,H), whe e ωis a
non i ial unc ion, ωH 0, and ω 0, om i s adjoin equa ion is iden ical o Eq. (2):
G∗
γϒω(,H)ϒ(,H,...)Gγ.
De ini ion 4 Equa ion (1) is said o be nonlinea ly sel -adjoin i he equa ion ob ained by subs i u ing ϒω(,H), whe e ω(,
H) is a nonze o unc ion o and H, om i s adjoin equa ion is iden ical o Eq. (2):
G∗
γϒω(,H)ϒ(,H,...)Gγ.
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3 Lie analysis
De ini ion 5 The ollowing is an ou line o he in ini esimal gene a o Mp olonga ion:
Msk(,υ,ι,H)∂k+ϑk(,υ,ι,H1)∂Hk+.. +ϑk1..ks(,υ,ι,H1,..,Hs)∂Hk1..ks.
In his ins ance, ϑscan be shown as ollows:
ϑγ
kAk(ψγ)−Hγ
lAk(l),
ϑγ
kl Al(ϑγ
k)−Hγ
k Al( ),
.
.
.
ϑγ
k1....ksAks(ϑγ
k1....ks−1)−Hγ
k1...ks−1Aks−1Aks(l),
since, he o al di e en ial ope a o gi en in Eq. (3) is shown by Aks.
3.1 In ini esimal gene a o s
The one-pa ame e Lie g oup co esponds o he o m’s in ini esimal ans o ma ion [31]:
¯+ε1+O(ε2),
¯υυ+ε2+O(ε2),
¯ιι+ε3+O(ε2),
¯
HH+εϑ +O(ε2),
whe e 1,2,3,andϑ ep esen independen and dependen a iables o in ini esimal ans o ma ion, espec i ely. The ec o
ield o Eq. (1) can be shown as he i h p olonga ion o M:
M1(,υ,ι,H)∂+2(,υ,ι,H)∂υ+3(,υ,ι,H)∂ι+ϑ(,υ,ι,H)∂H.
The in a iance o m o Min Eq. (1) becomes:
M[4]Hι +3HHι +3HιH +6Hιι +1Hυι +5Hυυ +4Hι +3Hυ +2H|Eq.(1)0,
whe e he ou h p olonga ion, M[4], can be cons uc ed in he ollowing manne :
M[4] M+ϑ∂H+ϑυ∂Hυ+ϑ∂H +ϑυ∂Hυ +ϑ∂H +ϑυυυ∂Hυυυ
+ϑι∂Hι +ϑυ∂Hυ +ϑυ∂Hυ.
The Lie algeb a o Eq. (1) is six-dimensional:
M1∂,M2∂ι,M3∂υ,M4∂H,M5υ∂,
M6(y1)∂H
z1
+(y2)∂ι
z1
+(y3)(∂)
z1
+(y4)(∂υ)
z1
,(7)
whe e y1−4ι25+ι2
3+13−245−3H5,y295ι,y335+3υ3,y495υ,z1
13−245.whe e ε1 is a undamen al pa ame e in Table 1aand1b.
3.2 Symme y g oup
In his phase, we will employ he Lie symme y g oups om he associa ed symme ies o p oduce some p ecise no el esponses
[32].As,we ake:
Ji:(,υ,ι,H)→(¯,¯υ,¯ι,¯
H), Ji o 1 ≤i≤6,
d
dε(¯,¯υ,¯ι,¯
H)(¯,¯υ,¯ι,¯
H) wi h ( ¯,¯υ,¯ι,¯
H)|ε0(,υ,ι,H),
whe e he pa ame e εis ex emely iny. Conside :
1H+2Hυ+3Hι+ϑH,
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Table 1 Commu a o able Table 1a:
[Mi,Mj]M1M2M3
M1000
M2000
M3000
M4000
M5000
M6−915M1
z1+(41512M2
z1−132M2
z1)315M2
z1
315M3
z1−M2
Table 1b:
[Mi,Mj]M4M5M6
M100
915M1
z1+(−41512M2
z1+132M2
z1)
M200
−315M2
z1
M300
315M3
z1+M2
M40M2313M3
z1+915M4
z1
M5−M20−1215M5
z1M5
M6−313M3
z1−915M4
z1
1215M5
z10
wi h he use o he in ini esimal gene a o s ϑ,1,2,and3. We acqui e he subsequen :
J1:(,υ,ι,H)→(+ε,υ,ι,H),
J2:(,υ,ι,H)→(,υ,ι+ε,H),
J3:(,υ,ι,H)→(,υ+ε,ι,H),
J4:(,υ,ι,H)→(,υ,ι,H+ε),
J5:(,υ,ι,H)→(+ευ,υ,ι,H),
J6:(,υ,ι,H)→(ez2ε,υez3ε,ιez3ε,H+R1).
(8)
whe e R1(4ι25ez3ε−ι2
3ez3ε−231ez2ε+445ez2ε−e−z2ε(4ε25−ι2
3−213+413+445+12 H5))
125,z235
z1,z3
95
z1. While J1,J2,andJ3 ep esen he ime in e p e a ion, and in a iance o space, espec i ely, in Eq. (8). The new solu ion Hi
should be ob ained using Jiwhe e 1 ≤i≤6, and we ha e a known solu ion o m H (,υ,ι):
H1 1(−ε,υ,ι),
H2 2(,υ,ι−ε),
H3 3(,υ−ε,ι),
H4e−ε 4(,υ,ι),
H5 5(,υ−ευ,ι),
H6e−ε 6((e−z2ε,υe−z3ε,ιe−z3ε)−R2).
whe e R2(4ι25ez3ε−ι2
3ez3ε−231ez2ε+445ez2ε−e−z2ε(4ε25−ι2
3−213+413+445))
125.
3.3 Op imal sys em
M{M1,M2,M3,M4}gene a es an abelian subalgeb a, as shown in Table 1a and b. Consequen ly, he one-dimensional op imal
sys em [33] is as ollows:
3.3.1 Una y class
Z1a<M1>,Z1b<M2>,Z1c<M3>,Z1d<M4>,Z1e<M5>,Z1 <M6>.
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489 Page 6 o 18 Eu . Phys. J. Plus (2024) 139:489
3.3.2 Bina y class
Z2al ;M1+c1M2>,Z2b<M1+c2M3>,Z2c<M1+c3M4>,Z2d<M1+c4M5>,
Z2dl ;M1+c5M6>,Z2e<M2+c6M3>,Z2 <M2+c7M4>,Z2g<M2+c8M5>,
Z2hl ;M2+c9M6>,Z2i<M3+c10 M4>,Z2j<M3+c11M5>,Z2k<M3+c12 M6>,
Z2ll ;M4+c13 M5>,Z2m<M4+c14 M6>,Z2n<M5+c15 M6>.
3.3.3 Te na y class
Z3al ;M1+aM2+bM3>,Z3b<M1+a1M2+b1M4>,Z3c<M1+a2M2+b2M5>,
Z3dl ;M1+a3M2+b3M6>,Z3e<M2+a4M3+b4M4>,Z3 <M2+a5M3+b5M5>,
Z3gl ;M2+a6M3+b6M6>,Z3h<M3+a7M4+b7M5>,Z3i<M3+a8M4+b8M6>,
Z3j<M4+a9M5+b9M36 >.
3.3.4 Combina ion o all classes
Z4<M1+d1M2+d2M3+d3M4>.
We calcula e simila i y a iables which a e used o calcula e all ob ainable simila i y educ ions o Eq. (1).
3.4 Reducing simila i y h ough una y class
Case 1: The Lag ange s uc u e co esponding o he ec o ield Z1a<∂
>is shown as ollows:
d
1dυ
0dι
0dH
0.
U ilizing he shown de ails, we asce ain he o m o he simila i y unc ion and simila i y a iables:
H(,υ,ι)R(Y,T), whe e Yυ,Tι,(9)
by subs i u ing Eq. (9) in o Eq. (1), we de i e he educed (1 + 1)-dimensional nonlinea PDE exp essed as:
6RTT +1RYT +5RYY 0.(10)
F om Eq. (10), we ob ain he subsequen explici solu ion o Eq. (1):
R(Y,T)G1−Y2
1−456−2T5+Y1
25+G2Y2
1−456+2T5−Y1
25, (11)
by ein oducing Eq. (11) in o Eq. (9), we a i e a he solu ion o Eq. (1):
H(,υ,ι)G1−υ2
1−456−2ι5+υ1
25+G2υ2
1−456+2ι5−υ1
25.
Case 2: The Lag ange con igu a ion co esponding o he ec o ield Z1b<∂
ι>is p o ided as ollows:
d
0dυ
0dι
1dH
0.
Wi h he p o ided in o ma ion, we de i e he simila i y unc ion and simila i y a iables in his manne :
H(,υ,ι)R(X,Y), whe e X,Yυ, (12)
by subs i u ing Eq. (12) in o Eq. (1), we de i e he educed (1 + 1)−dimensional nonlinea PDE shown as:
5RYY +3RYX +2RXX 0.(13)
F om Eq. (13), we ob ain he subsequen explici solu ion o Eq. (1):
R(Y,T)G3−X2
3−452−2Y2+X3
22+G4X2
3−452+2Y2−X3
22,(14)
by ein oducing Eq. (14) in o Eq. (12), we a i e a he solu ion o Eq. (1):
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Eu . Phys. J. Plus (2024) 139:489 Page 7 o 18 489
H(,υ,ι)G3−2
3−452−2υ2+3
22+G42
3−452+2υ2−3
22.
Case 3: The Lag ange amewo k co esponding o he ec o ield Z1c<∂
υ>is shown as ollows:
d
0dυ
1dι
0dH
0.
U ilizing he p o ided da a, we asce ain he s uc u e o he simila i y unc ion and simila i y a iables:
H(,υ,ι)R(X,T), whe e X,Tι, (15)
by subs i u ing Eq. (15) in o Eq. (1), we de i e he educed (1 + 1)−dimensional nonlinea PDE exp essed as:
RXXXT +3RXRXT +3RTRXX +6RTT +4RXT +2RXX 0.(16)
F om Eq. (16), we ob ain he subsequen explici solu ion o Eq. (1):
R(Y,T)2p2Tanhp2T(4p2
2−4+16p4
2+8p2
24−426+2
4)
26
+Xp2+p1.(17)
By ein oducing Eq. (17) in o Eq. (15), we a i e a he solu ion o Eq. (1):
H(,υ,ι)2p2Tanhc2ι(4p2
2−4+16c4
2+8p2
24−426+2
4)
26
+p2+p1.
Case 4: The Lag ange o ganized co esponding o he ec o ield Z1d<∂
H>is shown as ollows:
d
0dυ
0dι
0dH
1.
We de e mine he simila i y unc ion and simila i y a iables in he ollowing manne using he p o ided in o ma ion:
H(,υ,ι)R(X,Y), whe e X,Yυ, (18)
by subs i u ing Eq. (18) in o Eq. (1), we de i e he educed (1 + 1)-dimensional PDE shown as:
5RYY +3RYX +2RXX 0.(19)
F om Eq. (19),we ob ain he subsequen explici solu ion o Eq. (1):
R(X,Y)G5−X2
3−452−2Y2+X3
22+G6X2
3−452+2Y2−X3
22,(20)
by ein oducing Eq. (20) in o Eq. (18), we a i e a he solu ion o Eq. (1):
H(,υ,ι)G5−2
3−452−2υ2+3
22+G62
3−452+2υ2−3
22.
3.5 Reducing simila i y u ilizing bina y class
Case 5: The Lag ange s uc u e co esponding o he ec o ield Z2a<∂
+c1∂ι>is shown as ollows:
d
1dυ
0dι
c1dH
0.
We cons uc he simila i y unc ion and simila i y a iables in he ollowing app oach using ou supplied in o ma ion:
H(,υ,ι)R(η), ηι−c1. (21)
Upon subs i u ing he simila i y a iables Eq. (21) in o Eq. (1), we a i e a he subsequen ODE:
−c3
1R +3c2
1(R)2+(6−c14+c2
12)R0.(22)
Case 6: The Lag ange ounda ion co esponding o he ec o ield Z2b<∂
+c2∂υ>is shown as ollows:
d
1dυ
c2dι
0dH
0.
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489 Page 8 o 18 Eu . Phys. J. Plus (2024) 139:489
We es ima e he simila i y unc ion and simila i y a iables in he ollowing app oach using ou supplied in o ma ion:
H(,υ,ι)R(η), ηυ−c2, (23)
om Eq. (23), we ob ain he subsequen explici solu ion o Eq. (1):
R(η)g1+ηg2.(24)
Pu ing Eq. (24) in o Eq. (23), we a i e a he solu ion o Eq. (1):
H(,υ,ι)g1+(υ−c2)g2.
3.6 Reducing simila i y h ough e na y class
Case 7: The Lag ange o de co esponding o he ec o ield Z3a<∂
+a∂ι+b∂υ>is shown as ollows:
d
1dι
adυ
bdH
0.
We gene a e he simila i y unc ion and simila i y a iables in he ollowing app oach using he included in o ma ion:
H(,υ,ι)R(η), η−aι−bυ. (25)
The subsequen ODE is ob ained by inse ing simila i y a iables Eq. (25) in o Eq. (1):
−aR −6aRR +(6a2+ab1+b25−a4−b3+2)R 0.(26)
A e in eg a ing Eq. (26) and aking he cons an o in eg a ion equal o ze o, we can ge :
aR +3a(R)2−(6a2+ab1+b25−a4−b3+2)R0.(27)
3.7 Reducing simila i y u ilizing Combina ion o all classes
Case 8: The Lag ange s uc u e co esponding o he ec o ield Z4<∂
+d1∂ι+d2∂υ+d3∂H>is shown as ollows:
d
1dι
d1dυ
d2dH
d3
.
We compu e he simila i y unc ion and simila i y a iables in a speci ic manne using he shown in o ma ion:
H(,υ,ι)d3−R(η), ηd2ι−d1υ, (28)
om Eq. (28), we ob ain he subsequen explici solu ion o Eq. (1):
R(η)h1+ηh2.(29)
By ein oducing Eq. (29) in o Eq. (28), we a i e a he solu ion o Eq. (1):
H(,υ,ι)d3−h1−(d2ι−d1υ)h2.(30)
Rema k In his sec ion, we ha e de e mined he simila i y unc ion and simila i y a iables o Z1a,Z1b,Z1c,Z1d,Z2a,Z2b,Z3a,
and Z4. Al hough he simila i y unc ion and a iables o o he cases a e also compu able, we ha e chosen o exclude hem he e
o he sake o b e i y.
3.7.1 T a eling wa e solu ions om Eq. (27)
In his sec ion, we aim o de e mine he a eling wa e s uc u es o Eq. (1)usingEq.(27) h ough a no el ex ended di ec algeb aic
me hod. By equa ing he linea e m R in Eq. (27) and he nonlinea e m (R)2in Eq. (27), we a i e a a solu ion s uc u ed as
ollows:
R(η) 0+ 1g(η).(31)
Suppose g(η) ep esen s he solu ion o he subsequen equa ion:
g(η)ln(ρ)(1+2g(η)+3g2(η)), (32)
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Eu . Phys. J. Plus (2024) 139:489 Page 9 o 18 489
by subs i u ing Eqs. (31)and(32)in o(27) and hen equa ing he coe icien s o powe s o g(η), we deduce he subsequen sys em
o algeb aic equa ions:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
(g(η))0:31 12ln(ρ)212− 12ln(ρ)1−12 11ln(ρ)1+1 1ln(ρ)3221+21
1ln(ρ)3312−12 16ln(ρ)1−22 15ln(ρ)1+1 14ln(ρ)1+2 1
3ln(ρ)10,
(g(η))1:1 1ln(ρ)323− 12ln(ρ)2+1 14ln(ρ)2+2 13ln(ρ)2−12 16
ln(ρ)2+61 12ln(ρ)212−22 15ln(ρ)2+81 1ln(ρ)3312−12
11ln(ρ)20,
(g(η))2:31 12ln(ρ)222− 12ln(ρ)3−12 16ln(ρ)3+61 12ln(ρ)213−22
15ln(ρ)3+1 14ln(ρ)3+2 13ln(ρ)3+81 1ln(ρ)3321+71 1
ln(ρ)3322−12 11ln(ρ)30,
(g(η))3:61 12ln(ρ)223+121 1ln(ρ)33220,
(g(η))4:61 1ln(ρ)333+31 12ln(ρ)2320.
(33)
A e sol ing he desc ibed sys em wi h maple o ind 0, 1,anda, we de i e he subsequen se s o solu ions:
0 0, 1−23ln(ρ), a−B+√P+4
26
,
B16122
3ln(ρ)4−812
23ln(ρ)4+4
2ln(ρ)4+813ln(ρ)221−22
2ln(ρ)221
−813ln(ρ)24+22
2ln(ρ)24+2
22
1−42
256−2214+4236−426.
P16122
3ln(ρ)4−812
23ln(ρ)4+4
2ln(ρ)4+813ln(ρ)221−22
2ln(ρ)221
−813ln(ρ)24+22
2ln(ρ)24+2
22
1−42
256−2214+4236−426,
whe e ρ 0, 1, and 1,2,3,and 0a e cons an s. By making he assump ion 2
2−413, he solu ions o Eq. (32) can
be shown as ollows:
Family 1. When <0and3 0, hen:
H1(,υ,ι) 0−ln(ρ)−2+√− anρ(√−
2η),
H2(,υ,ι) 0−ln(ρ)−2−√−co ρ(√−
2η),
H3(,υ,ι) 0−ln(ρ)−2+√−( anρ(√−η)±√ ssecρ(√−η)),
H4(,υ,ι) 0−ln(ρ)−2−√−(co ρ(√−η)±√ scscρ(√−η)),
H5(,υ,ι) 0−ln(ρ)−2+√−
2 anρ√−
4η−co ρ√−
4η.
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489 Page 16 o 18 Eu . Phys. J. Plus (2024) 139:489
Family 2: Fo M2,weha eBγ
2−Hιand ι1, we acqui ed:
νιS−Hιϒυι +Hιιϒυ+Hιϒι+Hιιϒι,
ν−Hι[−4Hυϒ+ϒυι +ϒυ −42Hϒυ+4Hϒυ +4ϒ Hυ]+Hιιϒυ
−Hι[32ϒυ −ϒι−4ϒυH−1ϒHυ −4ϒυHυ]−ϒ2HHυιϒHιι
−4Hυι[4ϒH −4ϒυH−4(ϒ2Hυ +3υHH2Hυ +3ϒυHυ)],
νυ−Hι[4Hψ−1ϒυH −1ϒHυ +ϒυ +4Hϒυ+4HHυ
+4Hυϒυ+ϒHυ −ϒυυ]−Hι[−ϒυ −4ϒHυ −4ϒυH+ϒυ]
−H υ[−ϒ −ϒHυι +1ϒH −4ϒH −4ϒυH+ϒυ]−4ϒH
Hυι −ϒυυ −ϒυHυι·
Family 3: Fo M3,weha eBγ
3−Hυand υ1, we ob ained:
νιHυιϒυ+Hυϒι−Hυϒυι +Hιυϒι,
νHυ[−4Hυϒ+ϒυι +ϒυ −4Hϒυ+4Hϒυ +4ϒ Hυ]+Hιυ ϒυ
+Hυ[3ϒυ −ϒι−4ϒυH+1ϒHυ −4ϒυHυ]−4ϒHυHυ +ϒυ−ϒHυι
−Hυυ[4ϒH −4ϒυH−4(ϒ +3ϒυHυ +ϒυHυ)]
−4ϒHHυυ,
νυS−Hυ[4Hψ−1ϒυH −1ϒHυ +4Hϒυ+4HHυ +4Hυϒυ
+ϒHυ −ϒυυ]−Hυ[−ϒυ −4ϒHυ −4ϒυH+ϒυ]−Hυυ[−ϒ
+1ϒH −4ϒH −4ϒυH+ϒυ]−4ϒHHυυ −ϒυυ −ϒυHυ,
whe e ϒ(,υ,ι,H) (ι+υ,H)+g(ι−υ,H).
In his sec ion, we ha e igo ously compu ed he conse a ion laws o Bγ
1. I is no ewo hy ha he conse a ion laws o Bγ
2,
Bγ
3,Bγ
4,Bγ
5,andBγ
6a e also compu able, al hough hey ha e been omi ed om ou discussion o b e i y.
4.2 Local conse a ion laws
4.2.1 Mul iplie app oach
S ep1 : Le us conside a mul iplie o he o m:
πσπσ(,H,H1,H2,...,H ).
The condi ion ≤pis c ucial o ob aining he conse ed ec o s o Eq. (2).
S ep2 : These ec o s mus sa is y he condi ion ν(ν1,ν2,ν3,...νq)whe e qis he numbe o independen ec o s:
πσ(Eσ)Ak(νk).(38)
S ep3 : We de i e he subsequen de e mining sys em om he a ia ional de i a i e o Eq. (38):
δ
δHγ[πσ(Eσ)] 0, (39)
and sol ing Eq. (39) p o ides he equi ed mul iplie s.
4.3 Conse ed ec o s o Eq. (1)
In his sec ion, we ha ness he mul iplie echnique o es ablish he conse a ion laws o Eq. (1), a pi o al s ep in ou analysis. The
de e mining equa ion go e ning he mul iplie π(,υ,ι,H), as shown om Eq. (39), is pi o al o his pu pose:
δ
δHγ[π(Hι +3HHι +3HιH +6Hιι +1Hυι +5Hυυ +4Hι +3Hυ +2H)] 0, (40)
whe e he o al de i a i e A,Aι,andAυa e de ined as:
A∂+Hγ
∂Hγ+Hγ
l∂Hγ
l+Hγ
lm∂Hγ
lm +...,
Aι∂ι+Hγ
ι∂Hγ+Hγ
ιl∂Hγ
l+Hγ
ιlm∂Hγ
lm +...,
Aυ∂υ+Hγ
υ∂Hγ+Hγ
υl∂Hγ
l+Hγ
υlm∂Hγ
lm +...·
123

Eu . Phys. J. Plus (2024) 139:489 Page 17 o 18 489
Th ough igo ous e alua ion o Eq. (40) and me iculous u he calcula ions, we de i e he ollowing wo mul iplie s:
π1(,υ,ι,H)F,π2(,υ,ι,H)G,(41)
whe e
FF⎛
⎝−υ2
1−456−2ι5+υ1
25⎞
⎠GG⎛
⎝
υ2
1−456+2ι5−υ1
25⎞
⎠.
The conse ed ec o s a e ob ained om Eq. (41)andEq.(38).
Case 1: The conse a ion laws o Eq. (1) in ol e h ee componen s o conse ed ec o s, deno ed as νι,ν,andνυ.These
conse a ion laws can be exp essed in e ms o π1(,υ,ι,H)F:
νι
13
2HFH
 +3
2H2
H2
F−HF6+HιF6+HF4+HυF1+H F,
ν
1−3
2FH
ι −3
2FHH
+3
2HFH
ι−HF4+HF2+HυF3−H F,
νυ
1−1
2HF1+HF12−456+HυF5.
Case 2: The conse a ion laws pe aining o π2(,υ,ι,H)Gcan be shown as ollows:
νι
23
2HGH +3
2H2
G−HG6+HιG6+HG4+HυG1+HG,
ν
2−3
2HGH
ι −3
2GHH
+3
2HGH
ι−HG4+HG2+Hυ3G−HG,
νυ
3−1
2HG1−H12−456
2G+HυG5.
He e, i is impe a i e o unde sco e ha he unc ions Fand Ga e ega ded as a bi a y, wi h Fand Gdeno ing hei espec i e
i s -o de explici de i a i es. I is c ucial o emphasize ha he sel -adjoin ness app oach consis en ly yields a g ea e numbe o
conse a ion laws compa ed o he mul iplie app oach. Each symme y iden i ied h ough he sel -adjoin ness me hod con ibu es
dis inc conse a ion laws o he model unde examina ion. Local conse a ion laws desc ibe conse a ion locally a each poin ,
while nonlocal conse a ion laws desc ibe conse a ion o e an ex ended egion.
5 Conclusions
In conclusion, his s udy del ed in o he comple ely gene alized Hi o a–Sa suma–I o (cgHSI) equa ion h ough Lie analysis, aiming
o achie e se e al objec i es: de i ing one-dimensional conjugacy classes using Lie poin symme ies o he abelian algeb a o
he Lie g oup, leading o model educ ions wi h simila i y a iables. Va ious echniques we e employed o ob ain new soli a y
wa e solu ions and exac explici solu ions o he equa ion, including he use o hype bolic, igonome ic, and a ional unc ions.
Addi ionally, a hyb id me hod combining he Lie symme y me hod wi h he ex ended di ec algeb aic me hod was employed o
igo ously analyze exac and soli on solu ions. Conse a ion laws o he equa ion we e also compu ed, and hei classi ica ion
was achie ed using nonlinea sel -adjoin ness heo y. In e es ingly, a g ea e numbe o conse a ion laws we e ob ained using
nonlinea sel -adjoin ness compa ed o he mul iplie app oach. O e all, he esul s o his s udy enhance ou unde s anding o
he Hi o a–Sa suma–I o equa ion and pa e he way o u u e esea ch in he ield. The newly disco e ed exac solu ions hold
p omise o applica ions ac oss a ious ields, p o iding aluable insigh s o ma hema icians and physicis s and d i ing u he
explo a ion o nonlinea models. A enues o u u e esea ch could include u he explo a ion in o bi u ca ion analysis, chaos heo y
applica ions, and conduc ing sensi i i y analysis on he model unde conside a ion. These di ec ions could p o ide deepe insigh s
in o he dynamics and obus ness o he model, enhancing i s applicabili y and unde s anding wi hin he ield.
Acknowledgemen s This a icle has been p oduced wi h he inancial suppo o he Eu opean Union unde he REFRESH – Resea ch Excellence Fo
Region Sus ainabili y and High- ech Indus ies p ojec numbe CZ .10.03.01/00/22_003/0000048 ia he Ope a ional P og amme Jus T ansi ion. AliR
Ansa i is hank ul o he suppo o he Gul Uni e si y o Science and Technology and he Cen e o Applied Ma hema ics and Bioin o ma ics (CAMB)
unde p ojec code: ISG – 63.
Funding Open access unding p o ided by he Scien i ic and Technological Resea ch Council o Tü kiye (TÜB˙
ITAK).
Da a a ailibili y Da a sha ing does no apply o his a icle as no da ase s we e gene a ed o analyzed du ing he cu en s udy.
123
489 Page 18 o 18 Eu . Phys. J. Plus (2024) 139:489
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